The length of latus rectum of the conic passing through the origin and having the property that normal at each point (x, y) intersects the x-axis at  x+1,0 is:

Let px,y  be any point on the conic. ∴ slope of normal l=−dxdyEquation of normal to conic at px,y is Y−y=−dxdyX−x⇒−y=−dxdyX−x     (on x-axis Y = 0)⇒X=x+ydydx  Given  X=x+1  ⇒  ydydx=1ydy=dx⇒y22=x+C.......i   Since the conic passes through origin, x = 0, y = 0 in (i) we get C = 0∴ From  (i)  y2=2x Which is parabola having length of latus rectum  = 4a=2units